[EM] Arrow's theorem and cardinal voting systems

Richard Lung voting at ukscientists.com
Thu Jan 16 10:25:40 PST 2020

This is obviously a very learned summary that would require considerable 
study to do justice to, much more than my old head is capable of, quite 
apart from severe personal distresses, that have been over-whelming our 
Yet, just the other day, after a half century of amateur study, to my 
surprise, my naive physicists mathematical text crawled across the 
finishing line of publication. Only the last chapter is of direct 
interest to electoral mathematicians. It contains an explanation of how 
to conduct a two-dimensional election: FAB STV 2D.
The two dimensions are Representation and Arbitration. The latter graphs 
at 90 degrees (neutrally) to the former, and the count is conducted 
without disturbing the normal one-dimensional count that is FAB STV. But 
taken together as a complex variable the count is according to the rules 
of complex variables.

Richard Lung.

On 15/01/2020 20:57, Forest Simmons wrote:
> Just a couple of additional thoughts:
> Besides Arrow and Gibbard-Sattherwaite we have lots of criterion 
> incompatibility results from Woodall, and defensive strategy criteria 
> from Mike Ossipoff, Steve Eppley, et. al.. In particular we never 
> worried about Later No Help and later No Harm until Woodall came 
> along. Venzke and Benham picked up the torch and brought Woodall into 
> the EM Listserv discussion.
> Many EM contributors have clarified which combinations of various 
> criteria of more practical than academic stripe are compatible or not: 
> Participation, FBC, Precinct Summability, Chicken, etc.
> In particular, we now know through the work of Ossipoff, Venzke, 
> Benham, and others that the Chicken Defense and Burial Defense 
> (against CW burial) are incompatible in the presence of Plurality and 
> the the FBC, unless we allow an explicit approval cutoff or some other 
> strategic switch on the ballots.  Standard ordinal ballots are not 
> adequate for this even when truncation and equal rankings (including 
> equal top) are allowed.   A non-standard ballot that allows us to get 
> compatibility to all of these except the CC is MDDA(sc) which is 
> Majority Defeat Disqualification Approval with symmetric completion 
> below the approval cutoff.  This method also satisfies other basic 
> criteria such as Participation, Clone Independence, Mono-Raise, 
> Mono-Add, and IDPA, for example.
> In the context of the current discussion, the approval cutoff or some 
> equivalent strategic switch is essential for the compatibility of 
> chicken resistance and burial resistance..  No strategy, no 
> compatibility. So basically there is no decent method that is 
> resistant to both Burial of the CW and Chicken offensives.  (IRV is 
> chicken resistant and has a form of burial resistance, but routinely 
> buries the CW unless voters strategically betray their favorite to 
> save the CW. Furthermore, it fails mono-raise.)
> Before collaborative efforts of EM List members there was no known 
> clone independent, monotonic method for electing from the uncovered 
> set.  The closest thing was Copeland, which is clone dependent.
> Again the main point is that Arrow, and Gibbard-Satterthwaite are not 
> the "end of history" for election methods, just like the collapse of 
> the USSR was not the end of history as Fukuyama once proclaimed or 
> Thatcher's famous TINA "there is no alternative" (to capitalism).  
> Arrow and G-S give very valuable insights and help us avoid 
> cul-de-sacs, but they are not the last word in election methods 
> progress.  The "end of history" and TINA slogans are an excuse for 
> giving up prematurely for lack of imagination. We cannot allow Arrow 
> and G-S to become excuses for lack of imagination in Election 
> Methods.  What if Yee had given up before inventing the beautiful Yee 
> diagrams that constitute an Electo-Kaleidoscope for the study of 
> election methods analogous to the telesope and the electron microscope 
> in astronomy as instruments in other branches of knowledge?
> On Mon, Jan 13, 2020 at 3:32 PM Forest Simmons <fsimmons at pcc.edu 
> <mailto:fsimmons at pcc.edu>> wrote:
>     Rob,
>     Thanks for starting this great thread!
>     The "no perfect car" analogy is good.  More definite is the "no
>     100 percent efficient internal combustion engine" analogy that
>     follows from the second law of thermodynamics.  It applies to all
>     kinds of engines, but that doesn't mean that internal combustion
>     is as good as it gets.
>     If Gibbard-Satterthwaite tells us that we cannot have all of the
>     nice properties we want in one election method, that doesn't mean
>     that one method is as good as the next.
>     It follows from Arrow that we cannot have the Majority Criterion
>     and the IIAC at the same time, but there are many decent methods
>     (like River) that do satisfy the MC, and a bunch of other nice
>     properties, like Monotonicity, Clone Independence, the Condorcet
>     Criterion, and Independence from Pareto Dominated Alternatives, as
>     well as the basic Neutrality and Anonymity fairness criteria.
>     The way to think of Arrow's "Dictator" theorem is that it is
>     extremely hard to get a rankings based method with even minimal
>     decency conditions (like non-dictatorship) without scuttling the IIAC.
>     In other words, no decent ordinal based method can satisfy the
>     IIAC, which is the same point of view that Toby and Eppley
>     expressed.  It comes down to the mere existence of a Condorcet
>     Cycle.  Here's the subtle part that most people don't understand. 
>     Condorcet Cycles can exist in the preference schedules of an
>     election even if the election method makes no mention of
>     Condorcet, for example even in IRV/Hare/STV/RCV elections:
>     45 A>B>C
>     20 B>C>A
>     35 C>A>B
>     There exists a majority preference cycle A>B>C>A even though it
>     causes no problem for IRV, since B is eliminated and then C is the
>     majority winner between the two remaining candidates.
>     Now let's check the IIAC.  Suppose that A, one of the losers
>     withdraws from the race.  Then the winner changes from C to B,
>     since B beats C by a majority.  This shows that IRV does not
>     satisfy the IIAC, because removing a loser from the ballot changes
>     the winner.
>     But this is not just a problem for IRV, it's a problem for any
>     method that respects the Majority Criterion; if the method makes A
>     the winner, then removing B changes the winner.  If it makes B the
>     winner, then removing C changes the winner.  If it makes C the
>     winner, then (as we saw in the case of IRV above) removing A
>     changes the winner. to B.
>     So Arrow's "paradox" can be considered as forcing us to realize
>     that the IIAC is not a realistic possibility in the presence of
>     ordinal ballots because such ballots allow us to detect oairwise
>     (head-to-head) preferences, and when it comes down to a single
>     pair of candidates the Majority Criterion says the pairwise winner
>     must be chosen,
>     However, as someone mentioned, Approval Voting avoids this
>     "paradox" once the ballots have been submitted, since the Approval
>     winner A is always the "ballot CW," and in two different ways:(1)
>     For any other candidate X, candidate A will be rated above X on
>     more ballots than not, and (2) A's approval score will be higher
>     than the sore of any other candidate.  From either point of view,
>     if we remove a loser Y from the ballots, then A will still be the
>     winner according to the same ballots with Y crossed out.
>     That's at the ballot level.  But if Y withdrew before the ballots
>     were filled out, it could change the winner, because if Y were the
>     only approved candidate for a certain voter before the withdrawal,
>     that voter might decide to lower her personal approval cutoff
>     before submitting her ballot.  Or she could raise the cutoff if Y
>     had been the only disapproved candidate.
>     As others have mentioned in this discussion, Approval Voting
>     externalizes the problem of the IIAC from being a decision problem
>     for the method itself to a strategical decision problem for the
>     voter. A voter might think of that as an unfair burden.
>     One answer to this problem could be DSV (Designated Strategy
>     Voting): You submit your sincere ratings, and the DSV machine
>     applies a strategy of your choice or a default strategy to
>     transform the ballots into approval style ballots.  Rob LeGrand
>     explored some of the possibilities and limitations of this
>     approach in his master's thesis.  He doesn't claim to have
>     exhausted the possibilities.  (I also have some ideas in this vein
>     that still need exploring.)
>     What constitutes a "sincere rating."  One approach to that has
>     already been mentioned in the ice-cream flavor context in this
>     thread. Another is to use as a rating for candidate X your
>     subjective probability that on a typical issue of any significance
>     candidate X would support the same side you support.
>     It's not just Approval that requires some hard thinking in
>     conjunction with filling out the ballots. Ranking many candidates
>     (think about the number of candidates in the election that
>     propelled Schwarznegger into office) may be just as burdensome as
>     trying to decide exactly which candidates to mark as approved. In
>     Australia you can get around this difficulty by copying "candidate
>     cards" or by voting the party line. Presumably these experts are
>     reflecting state of the art strategy in their rankings ... the
>     strategy that is indispensable for optimum results according to
>     Gibbard-Satterthwaite.  This is not just a problem of Approval,
>     though it may seem worse in Approval.  In actuality, aoproval and
>     score/range are the only commonly used methods where optimal
>     strategy never requires you to "betray " your favorite.
>     To cut the Gordian knot of this complexity Charles Dodgson (aka
>     Lewis Carroll) suggested what we now call Asset Voting. Each Voter
>     delegates her vote to the candidate she trusts the most to
>     rep[resent her in the decision process. Since write-ins are
>     allowed, she can write in herself if she doesn't trust anybody
>     else to be her proxy.  These proxies get together with their
>     "assets"  (delegated votes) and choose a winner by use of some
>     version of Robert's Rules of Order.
>     Which criteria are satisfied by this method? Does Gibbard
>     Satthethwaite have anything to say about it? How about Arrow?  For
>     that matter does first past the post plurality satisfy the IIAC?
>     (No more or less than Approval in reality.)
>     Let's talk about Gibbard-Satterthwaite.  Is there any incentive
>     for a person to delegate as proxy someone other than her favorite?
>     If we are talking representative democracy, then why would you
>     want to delegate your vote to candidate B when candidate A was the
>     one you trusted most to represent you in making important
>     decisions once in office?
>     All of the "problems" with the method are essentially externalized
>     to the deliberations governed by"Robert's Rules of Order" in the
>     smoke filled room.
>     Gibbard-Satterthwaite is taken to say that it is impossible to
>     obtain sincere preferences or sincere utilities from voters in the
>     context of full information (or disinformation) elections.  Yet it
>     turns out to be relatively easy; you just need to separate the
>     ballot into two parts.  The first part requires strategic voting
>     to pick the two alternatives as finalists.  The second part is
>     used solely to choose between these two options.  (In the case of
>     cardinal ballots the finalists are lotteries.)  A version of the
>     uncertainty principle obtains here: if you use the sincere ballots
>     for any other instrumental purpose than to choose between the two
>     finalists, then you almost certainly destroy their sincerity.
>     However there would be no problem comparing tthe sincere part with
>     the strategical part to get statistics about voters' willingness
>     to vote insincerely in choosing the finalists.
>     Another avenue that has been barely explored is the use of chance
>     to incentivize consensus when there is a potential for it.
>     For example, suppose that preferences are
>     60 A>C>>>>B
>     40 B>C>>>>A
>     Under approval voting the A faction has a strong incentive to
>     downgrade C and vote 60 A>>>>>C>B making A the (insincere)
>     approval winner as Gibbard-satterthwaite would predict.
>     However, if the rules said that in the absence of a full consensus
>     approval winner, the winner would be chosen by random ballot, then
>     (assuming rational voters voting in their own interest) C would be
>     the sure outcome; no rational voter in either faction would prefer
>     random ballot expectations over a sure deal on C.
>     Jobst Heitzig is the pioneer in this area.
>     In sum, Arrow, et.al <http://et.al>. should not constitute a nail
>     in the coffin of creative progress in Election Methods.  IMHO that
>     is an important message we need to send if we want to attract new
>     talent.
>     Forest
> ----
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